Operational applications of the diamond norm and related measures in quantifying the non-physicality of quantum maps

Bartosz Regula; Ryuji Takagi; Mile Gu

doi:10.22331/q-2021-08-09-522

Operational applications of the diamond norm and related measures in quantifying the non-physicality of quantum maps

Bartosz Regula¹, Ryuji Takagi¹, and Mile Gu^1,2,3

¹Nanyang Quantum Hub, School of Physical and Mathematical Sciences, Nanyang Technological University, 637371, Singapore
²Complexity Institute, Nanyang Technological University, 637371, Singapore
³Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543, Singapore

Published:	2021-08-09, volume 5, page 522
Eprint:	arXiv:2102.07773v3
Doi:	https://doi.org/10.22331/q-2021-08-09-522
Citation:	Quantum 5, 522 (2021).

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Abstract

Although quantum channels underlie the dynamics of quantum states, maps which are not physical channels — that is, not completely positive — can often be encountered in settings such as entanglement detection, non-Markovian quantum dynamics, or error mitigation. We introduce an operational approach to the quantitative study of the non-physicality of linear maps based on different ways to approximate a given linear map with quantum channels. Our first measure directly quantifies the cost of simulating a given map using physically implementable quantum channels, shifting the difficulty in simulating unphysical dynamics onto the task of simulating linear combinations of quantum states. Our second measure benchmarks the quantitative advantages that a non-completely-positive map can provide in discrimination-based quantum games. Notably, we show that for any trace-preserving map, the quantities both reduce to a fundamental distance measure: the diamond norm, thus endowing this norm with new operational meanings in the characterisation of linear maps. We discuss applications of our results to structural physical approximations of positive maps, quantification of non-Markovianity, and bounding the cost of error mitigation.

Featured image: A schematic representation of decomposing a non-completely-positive map $\Phi$ into two completely positive trace-non-increasing (CPTNI) maps.

► BibTeX data

@article{Regula2021operational,
  doi = {10.22331/q-2021-08-09-522},
  url = {https://doi.org/10.22331/q-2021-08-09-522},
  title = {Operational applications of the diamond norm and related measures in quantifying the non-physicality of quantum maps},
  author = {Regula, Bartosz and Takagi, Ryuji and Gu, Mile},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {522},
  month = aug,
  year = {2021}
}

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